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Mirrors > Home > MPE Home > Th. List > subcss1 | Structured version Visualization version GIF version |
Description: The objects of a subcategory are a subset of the objects of the original. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
subcss1.1 | ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) |
subcss1.2 | ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) |
subcss1.b | ⊢ 𝐵 = (Base‘𝐶) |
Ref | Expression |
---|---|
subcss1 | ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subcss1.2 | . 2 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) | |
2 | eqid 2818 | . . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
3 | subcss1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
4 | 2, 3 | homffn 16951 | . . 3 ⊢ (Homf ‘𝐶) Fn (𝐵 × 𝐵) |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → (Homf ‘𝐶) Fn (𝐵 × 𝐵)) |
6 | subcss1.1 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) | |
7 | 6, 2 | subcssc 17098 | . 2 ⊢ (𝜑 → 𝐽 ⊆cat (Homf ‘𝐶)) |
8 | 1, 5, 7 | ssc1 17079 | 1 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 × cxp 5546 Fn wfn 6343 ‘cfv 6348 Basecbs 16471 Homf chomf 16925 Subcatcsubc 17067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-pm 8398 df-ixp 8450 df-homf 16929 df-ssc 17068 df-subc 17070 |
This theorem is referenced by: subcss2 17101 subccatid 17104 subsubc 17111 funcres 17154 funcres2b 17155 funcres2 17156 idfusubc 44065 |
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